If the punto was a tatty black 3 door with aftermarket alloys then yes I can see why you would walk away.
You have two options right now. You can get another old punto or similar for £2k which is already 10 - 14 years old, this might be alright but might not have been well looked after, may have the beginings of rust and may only last another 2-5 years before needing replacement again. This would fit the suggestion of a short term car to see you till you can get something better.
Or you could look and see what deals there are about to get something you actually want. You earn money, you own property, if you don't have the cash right now for something more expensive, why not look at a loan to get something better, which will last you.
As an example there is a Fiat Dealer in Colchester at the moment with a 2020, fiat Tipo with ~18,000 miles on the clock, advertised for about £10-11k Low spec but low miles and very new, or s slightly older but very nicely specced Fiat 500X with the same dealer group, heated seats, parking sensors, 2016 with 50ishk miles for under £10k. Depending on the duration and how much deposit you put down, a £10k car could cost under £200 a month, for a few years, after that it's yours; and say for example you took a 2020 Tipo and you looked after it well, that could easily last you till 2035 given your Punto is 15 years old now, were as a 2010 Punto Evo might only have another 5 years left in it or less if its not been looked after and many of these have not been well looked after. You'd be looking at buying another one in 5 years time, which could by that point be another car.
When it comes to borrowing, there are a lot of things to consider and you should seek advice but, in your position I would not hesitate to go and get a car on a PCP deal for 3 or 4 years, put down what ever I have as a deposit at the moment and then save up to either pay it off early or on a PCP pay the balloon payment off at the end.
Just make sure you get Gap insurance if you go the finance route.